Euclid's First Theorem (thing) by tongpoo

Mathematics > Number Theory

Sometimes called "Euclid's lemma" in textbooks when appearing before a proof of the fundamental theorem of arithmetic. It states that if p is a prime number and p|ab, then either p|a or p|b ("|" means "divides").

Corollaries:

Incidentally, Euclid's Second Theorem states that there are infinitely many primes.

References:
"Euclid's First Theorem" is sometimes referred as such according to MathWorld.com. Many of these theorems appear in Euclid's Elements. Book VII, proposition 30 states Euclid's First Theorem. Book IX, proposition 14 partially states the fundamental theorem of arithmetic, and proposition 20 states Euclid's Second Theorem. Thanks to Swap for tipping me off about this.

Proof that there is no largest prime number	The square root of any positive integer is either integral or irrational	Euclid's Elements: Book IX	Euclid's Elements: Book VII
Rational Root Theorem	divides	relatively prime	Euclid's lemma
Furstenberg's proof that there are infinitely many prime numbers	The first six books of the elements of Euclid	Simplex	Fermat's little theorem
Fundamental theorem of arithmetic	prime number